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  "body": "\n## Summary\n\nThe Gegenbauer transform is a finite [[integral transform|YCK709QB06]] on $[-1, 1]$.\nIt sends a function to its integrals against the [[Gegenbauer polynomials|5TRFA6F79B]] $C_n^{\\lambda}$ with the weight $(1 - x^2)^{\\lambda - 1/2}$.\nIt is the symmetric case $\\alpha = \\beta$ of the [[Jacobi transform|72TSY1Z4VA]], up to constant factors.\nIt turns the Gegenbauer differential operator into multiplication by $-n(n + 2\\lambda)$.\nFor $\\lambda = 1/2$ it is the [[Legendre transform|51PH7Z03TJ]].\n\n## Explanations\n\n### L1 human\n\nGegenbauer polynomials are orthogonal polynomials on $[-1, 1]$ for a weight that is symmetric about $0$.\nThey appear naturally for functions on spheres that only depend on one angle.\nThe Gegenbauer transform lists how much of each polynomial a function contains.\nFor example, with $\\lambda = 1$ the polynomials are the Chebyshev polynomials of the second kind, and the weight is $\\sqrt{1 - x^2}$.\n",
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